with a positive constant \(C\)  independent of \(\delta \)  and \(m\) . Note that \(J(\bar{u}(\cdot ))\)  can be replaced with \(\inf _{u(\cdot )}J(u(\cdot ))\)  if no optimal control exists. In this case the \(u^{m}(\cdot )\)  satisfies (REF ) is called a \(\delta ^{\frac{1}{2}}\) -optimal control that has been studied in [1]}.
Thus (REF ) also provides an approach to find the near-optimal control, and illustrates near-optimal controls are indeed more available than optimal ones as mentioned in [1]}.
